Question

The following game is played. The player randomly draws from the set of integers \(\\{1,2, \ldots, 20\\} .\) Let \(x\) denote the number drawn. Next the player draws at random from the set \(\\{x, \ldots, 25\\}\). If on this second draw, he draws a number greater than 21 he wins; otherwise, he loses. (a) Determine the sum that gives the probability that the player wins. (b) Write and run a line of \(\mathrm{R}\) code that computes the probability that the player wins. (c) Write an \(\mathrm{R}\) function that simulates the game and returns whether or not the player wins. (d) Do 10,000 simulations of your program in Part (c). Obtain the estimate and confidence interval, (1.4.7), for the probability that the player wins. Does your interval trap the true probability?

Short answer

The sum that gives the probability that the player wins is approximately 2.926. The R code to compute this probability is: win_prob <- sum(4/(25-1:21))/20. An R function, simulate_game, is created to simulate the game and return whether the player wins. After running 10,000 simulations of the game, we get the estimate and confidence interval for the winning probability.

Step-by-step solution

Determine the probabilities of winning

(a) To determine the sum of probabilities, we must recognize that the player can only win if they draw 22, 23, 24 or 25 in the second round. These numbers can only be drawn if the first draw is 21 or less. Hence, we sum the probabilities of drawing 22, 23, 24 or 25 after having drawn x from the first set. This results in \(\sum_{x=1}^{21} \frac{4}{25-x} = \frac{4}{4} + \frac{4}{5} + \ldots + \frac{4}{24} = 1 + 0.8 + \ldots + 0.167 = 2.926\).

Writing R code to compute the winning probability

(b) After getting the sum, divide by the total number of possible outcomes (20) to get a probability. Write a line of R code for this: win_prob <- sum(4/(25-1:21))/20

Writing an R function to simulate the game

(c) Now we define an R function to simulate the game. The function draws a number from 1:20, then draws a second number from the first number to 25 and checks if it is greater than 21: simulate_game <- function() { first_draw <- sample(1:20, 1); second_draw <- sample(first_draw:25, 1); return(second_draw > 21) }

Running simulations to estimate and find the confidence interval of the winning probability

(d) Now that we have a game simulation function, we simulate the game 10,000 times and calculate the winning probability and the confidence interval. The R code for this is: simulation_results <- replicate(10000, simulate_game()); win_estimate <- sum(simulation_results)/10000; conf_interval <- qnorm(0.975)*sqrt(win_estimate*(1-win_estimate)/10000); paste('[', win_estimate - conf_interval, ', ', win_estimate + conf_interval, ']')